Second Preimages on n-bit Hash Functions for Much Less than 2n Work
John Kelsey1 and Bruce Schneier2
1 National Institute of Standards and Technology, [email protected] 2 Counterpane Internet Security, Inc., [email protected]
Abstract. We expand a previous result of Dean [Dea99] to provide a second preimage attack on all n-bit iterated hash functions with Damg ard- Merkle strengthening and n-bit intermediate states, allowing a second preimage to be found for a 2k-message-block message with about k 2n/2+1 +2n k+1 work. Using RIPEMD-160 as an example, our attack can nd a second preimage for a 260 byte message in about 2106 work, rather than the previously expected 2160 work. We also provide slightly cheaper ways to nd multicollisions than the method of Joux [Jou04]. Both of these results are based on expandable messages–patterns for producing messages of varying length, which all collide on the intermediate hash re- sult immediately after processing the message. We provide an algorithm for nding expandable messages for any n-bit hash function built using the Damg ard-Merkleconstruction, which requires only a small multiple of the work done to nd a single collision in the hash function.
1 Introduction
The security goal for an n-bit hash function is that collisions require about 2n/2 work, while preimages and second preimages require about 2n work. In [Dea99], Dean demonstrated that this goal could not be accomplished by hash functions whose compression functions allowed the easy nding of xed points, such as MD5 [Riv92] and SHA1 [SHA02]. In this paper, we use the multicollision- nding result of [Jou04] to demonstrate that the standard way of constructing iterated hash functions (the Damg ard-Merkleconstruction) cannot meet this goal, regardless of the compression function used. Thus, hash functions such as RIPEMD-160 [DBP96] and Whirlpool [BR00] (when used with a full 512- bit result) provide less than the previously-expected amount of resistance to second-preimage attacks, just as do hash functions like SHA1. For a message of 2k message blocks, we provide a second preimage attack requiring about k 2n/2+1 + 2n k+1 work. Like Dean’s attack, ours is made possible by the notion of expandable messages–patterns of messages of di erent lengths which all yield the same intermediate hash value after processing them. These expandable messages do not directly yield collisions on the whole hash function because of the length padding done at the end of modern hash func- tions, and in any event are no easier to nd than collisions. However, they allow second preimages and multicollisions to be found much more cheaply than had previously been expected. This result may be compared with an earlier generic preimage attack by Merkle: the attacker is given 2k distinct n-bit hash outputs, and expects to nd a preimage for one of the outputs with about 2n k work, but has no ability to choose which of the outputs is to be matched. (Note that Merkle’s attack is truly generic, in that it applies to any hash fuction with an n-bit output, even a random oracle.) Our attack, like the earlier attack of Dean, probably has no practical impact on the security of any system currently relying upon a hash function such as SHA1, Whirlpool, or RIPEMD-160. This is true because the attack is always at least as expensive as collision search on the hash function, and because the di culty of the attack grows quickly as the message gets shorter. For example, a 160-bit hash function like SHA1 or RIPEMD-160 requires about 2128 work to nd a second preimage for a 238-byte message, and a target message of only one megabyte (220 bytes) requires about 2146 work to nd a second preimage. Also, the attack only recovers second preimages–it doesn’t allow an attacker to invert the hash function. The signi cance of our result is in demonstrating another important way in which the behavior of the hash functions we know how to construct di ers from both the commonly claimed security bounds of these functions, and from the random oracles with which we often model them. When combined with the recent results of Joux [Jou04], our results raise questions about the usefulness of the widely-used Damg ard-Merkleconstruction for hash functions where attackers can do more than 2n/2 work. The remainder of the paper is organized as follows: First, we discuss basic hash function constructions and security requirements. Next, we demonstrate a generic way to nd expandable messages, and review the method of Dean. We then demonstrate how these expandable messages can be used to violate the second preimage resistance of nearly all currently speci ed cryptographic hash functions with less than 2n work. Finally, we demonstrate an even more e cient (albeit much less elegant) way to nd multicollisions than the method of Joux, using Dean’s xed-point-based expandable messages. We end with a discussion of how this a ects our understanding of iterated hash function security.
2 Hash Function Basics
In 1989, Merkle and Damg ard[Mer89,Dam89] independently provided security proofs for the basic construction used for almost all modern cryptographic hash functions3. Here, we describe this construction4, and its normal security claims.
3 These functions date back to Rabin, and were widely used by hash function designers throughout the 1980s [Pre05,MvOV96]. 4 Damg ardproposed two methods for constructing hash functions. This paper ad- dresses only the more commonly used one, which was independently invented by Merkle. A hash function with an n-bit output is expected to have three minimal security properties. (In practice, a number of other properties are expected, as well.) 1. Collision-resistance: An attacker should not be able to nd a pair of messages M 6= M 0 such that hash(M) = hash(M 0) with less than about 2n/2 work. 2. Preimage-resistance: An attacker given an output value Y in the range of hash should not be able to nd an input X from its domain so that Y = hash(X) with less than about 2n work. 3. Second preimage-resistance: An attacker given one message M should not be able to nd a second message, M 0 to satisfy hash(M) = hash(M 0) with less than about 2n work. A collision attack on an n-bit hash function with less than 2n/2 work, or a preimage or second preimage attack with less than 2n work, is formally a break of the hash function. Whether the break poses a practical threat to systems using the hash function depends on speci cs of the attack. Following the Damg ard-Merkleconstruction, an iterated hash function is built from a xed-length component called a compression function, which takes an n-bit input chaining value and an m-bit message block, and derives a new n-bit output chaining value. In this paper, F (H,M) is used to represent the ap- plication of this compression function on hash chaining variable H and message block M. In order to hash a full message, the following steps are carried out: 1. The input string is padded to ensure that it is an integer multiple of m bits in length, and that the length of the original, unpadded message appears in the last block of the padded message. 2. The hash chaining value h[i] is started at some xed IV, h[ 1], for the hash function, and updated for each successive message block M[i] as h[i] = F (h[i 1],M[i]) 3. The value of h[i] after processing the last block of the padded message is the hash output value. This construction gives a reduction proof: If an attacker can nd a collision in the whole hash, then he can likewise nd one in the compression function. The inclusion of the length at the end of the message is important for this reduction proof, and is also important for preventing a number of attacks, including long- message attacks [MvOV96]. Besides the claimed security bounds, there are two concepts from this brief discussion that are important for the rest of this paper: 1. A message made up of many blocks, M[0, 1, 2, ..., 2k 1], has a corresponding sequence of intermediate hash values, h[0, 1, 2, ..., 2k 1]. 2. The padding of the nal block includes the length, and thus prevents colli- sions between messages of di erent lengths in the intermediate hash states from yielding collisions in the full hash function. 3 Finding Expandable Messages
An expandable message is a kind of multicollision, in which the colliding messages have di erent lengths, and the message hashes collide in the input to the last compression function computation, before the length of the message is processed. Consider a starting hash value h[ 1]. Then an “expandable message” from h[ 1] is a pattern for generating messages of di erent lengths, all of which yield the same intermediate hash value when they are processed by the hash, starting from h[ 1], without the nal padding block with the message length being included. In the remainder of the paper, an expandable message that can take on any length between a and b message blocks, inclusive, will be called an (a, b)-expandable message.
3.1 Dean’s Fixed-Point Expandable Messages
In [Dea99], there appears a technique for building expandable messages when xed points can easily be found in the compression function5. For a compression function h[i] = F (h[i 1],M[i]), a xed point is a pair (h[i 1],M[i]) such that h[i 1] = F (h[i 1],M[i]). Compression functions based on the Davies-Meyer construction [MvOV96], such as the SHA family [SHA02], MD4, MD5 [Riv92], and Tiger [AB96], have easily found xed points. Similarly, Snefru [Mer90] has easily found xed points. Techniques for nding these xed points for com- pression functions based on the Davies-Meyer construction appear in [MOI91], and are brie y discussed in an appendix to this paper, along with techniques for nding xed points in Snefru. Note that these techniques produce a pair (h[i 1],M[i]), but allow no control over the value of h[i 1]. We can construct an expandable message using xed points for about twice as much work as is required to nd a collision in the hash function. This is done by rst nding about 2n/2 randomly-selected xed points for the compression function, and then trying rst message blocks until one leads from the initial hash value to one of the xed points. ALGORITHM: MakeFixedPointExpandableMessage(h[in]) Make an expandable message from initial hash value h[in], using a xed point nding algorithm.
Variables: 1. h[in] = initial chaining value for the expandable messages. 2. FindRandomFixedPoint() = an algorithm returning a pair (h[i],M[i]) such that h[i] = F (h[i],M[i]). 3. A, C = two lists of hash values. 4. B,D = two lists of message blocks. 5. i, j = integers.
5 We were made aware of Dean’s work by a comment from one of the anonymous referees. 6. M(i) = a function that produces a unique message block for each integer i less than 2n. 7. n = width of hash function chaining value and output. Steps: 1. Construct a list of 2n/2 xed points: – For i = 0 to 2n/2 1: h, m = FindRandomFixedPoint() A[i] = h B[i] = m 2. Construct a list of 2n/2 hash values we can reach from h[ 1]: – For i = 0 to 2n/2 1: h = F (h[in],M(i)) C[i] = h D[i] = M(i) 3. Find a match between lists A and C; let i, j satisfy A[i] = C[j]. 4. Return expandable message (D[j],B[i]). Work: About 2n/2+1 compression function computations, assuming 2n/2+1 memory.
If an n-bit hash function has a maximum of 2k blocks in its messages, then this technique takes about 2n/2+1 work to discover (1, 2k)-expandable messages. Producing a message of the desired length is trivial, consisting of one copy of the starting message block, and as many copies of the xed-point message block as necessary to get a full message of the desired length. ALGORITHM: ProduceMessageFP(R,X,Y ) Produce a message of desired length from the xed-point expandable mes- sages.
Variables: 1. R = the desired length in message blocks; must be at least one and no more than the maximum number of message blocks supported by the hash. 2. X = the rst message block in the expandable message. 3. Y = the second (repeatable) block in the expandable message. Steps: 1. M = X. 2. For i = 0 to R 2: – M = M||Y 3. Return M. Work: Negligible work, about R steps. 3.2 A Generic Technique: Multicollisions of Di erent Lengths Finding an expandable message for any compression function with n-bit inter- mediate hash values takes only a little more work than nding a collision in the hash function. This technique is closely related to the technique for nding k-collisions in iterated hash functions from Joux. In Joux’s technique, a sequence of single-message-block collisions is found, and then pasted together to provide a large number of di erent messages of equal length that lead to the same hash value. In our technique, a sequence of collisions between messages of di erent lengths is found, and pasted together to provide a set of messages that can take on a wide range of di erent lengths without changing the resulting intermediate hash value—an expandable message.
Finding a Collision on Two Messages of Di erent Lengths. Finding an expandable message requires the ability to nd many pairs of messages of di erent speci ed lengths that have the same resulting intermediate hash value. Finding such a pair is not fundamentally di erent than nding a pair of equal- length messages that collide: The attacker who wants a collision between a one- block message and an -block message constructs about 2n/2 messages of length 1, and about the same number of length , and looks for a collision. For e ciency, the attacker chooses a set of -block messages whose hashes can be computed about as e ciently as the same number of single-block messages.
ALGORITHM: FindCollision( , hin) Find a collision pair with lengths 1 and blocks, starting from hin. Variables: 1. = desired length of second message. 2. A, B = lists of intermediate hash values. 3. q = a xed “dummy” message used for getting the desired length. 4. hin = the input hash value for the collision. 5. htmp = intermediate hash value used in the attack. 6. M(i) = the ith distinct message block used in the attack. 7. n = width of hash function chaining value and output in bits. Steps: 1. Compute the starting hash for the -block message by processing 1 dummy message blocks: – htmp = hin. – For i = 0 to 2: htmp = F (htmp, q) 2. Build lists A and B as follows: – for i = 0 to 2n/2 1: A[i] = F (hin,M(i)) B[i] = F (htmp,M(i)) 3. Find i, j such that A[i] = B[j] 4. Return colliding messages (M(i), q||q||...||q||M(j)), and the resulting in- termediate hash F (hin,M(i)). Work: 1 + 2n/2+1 compression function calls Building a Full (k, k + 2k 1)-expandable message. We can use the above algorithm to construct expandable messages that cover a huge range of possible lengths, in a technique derived from the multicollision- nding technique of [Jou04]. We rst nd a colliding pair of messages, where one is of one block, and the other of 2k 1 +1 blocks. Next, we nd a collision pair of length either 1 or 2k 2 + 1, then 1 or 2k 3 + 1, and so on, until we reach a collision pair of length 1 or length 2. The result is a list of pairs of message components of di erent lengths, which lead to the same intermediate hash after processing them. The rst such pair allows a choice of adding 2k 1 blocks to the expanded message, the second allows a choice of adding 2k 2 blocks, and so on. Thus, expanding the message is just writing the di erence between the desired length and the number of message components in binary, and using each bit in that binary string to choose the corresponding short or long message component to include.
ALGORITHM: MakeExpandableMessage(hin, k) Make a (k, k + 2k 1)-expandable message.
Variables: 1. htmp = the current intermediate hash value. 2. C = a list of pairs of messages of di erent lengths; C[i][0] is the rst message of pair i, while C[i][1] is that pair’s second message. Steps: 1. Let htmp = hin. 2. For i = 0 to k 1: i – (m0, m1, htmp) = FindCollision(2 + 1, htmp) – C[k i 1][0] = m0 – C[k i 1][1] = m1 3. Return the list of message pairs C. Work: k 2n/2+1 + 2k k 2n/2+1 compression function calls.
At the end of this process, we have an k 2 array of messages, for which we have done approximately 2k + k 2n/2+1 compression function computations, and with which we can build a message consisting of between k and k + 2k 1 blocks, inclusive, without changing the result of hashing the message until the nal padding block.
Producing a Message of Desired Length. Finally, there is a simple algo- rithm for producing a message of desired length from an expanded message. This amounts to simply including the di erent-length pieces based on the bit pattern of the desired length. ALGORITHM: ProduceMessage(C, k, L) Produce a message of length L, if possible, from the expandable message speci ed by (C, k).
Variables: 1. L = desired message length. 2. k = parameter specifying that C contains a (k, k + 2k 1)-expandable message. 3. C = a k 2 array of message fragments of di erent lengths. 4. M = the message to be constructed. 5. T = a temporary variable holding the remaining length to be added. 6. S[0..k 1] = a sequence of bits from T . 7. i = an integer counter. Steps: 1. Start with an empty message M = ∅. 2. If L > 2k + k 1 or L < k, return an error condition. 3. Let T = L k. 4. Let S = the bit sequence of T , from low-order to high-order bits. 5. Concatenate message fragments from the expandable message together until we get the desired message length. Note that this is very similar to writing T in binary. – for i = 0 to k 1: if S[i] = 0 then M = M||C[i][0] else M = M||C[i][1] 6. Return M. Work: Negligible (about k table lookups and string copying operations).
The result of this is a message of the desired length, with the same hash result before the nal padding block is processed as all the other messages that can be produced from this expandable message.
3.3 Variants The expandable messages found by both of these methods can start at any given hash chaining value. As a result, we can build expandable messages with many useful properties:
1. The expandable message can start with any desired pre x. 2. The expandable message can end with any desired su x. 3. While both algorithms given here for nding expandable messages assume complete freedom over choice of message block, a variant of the generic method can be used even if the attacker is restricted to only two possible values for each message block. 4. The xed-point method requires about 2n/2 possible values for each message block, but this is su ciently exible that for existing hash functions, it can typically be used with only ASCII text, legitimate sequences of Pentium opcodes, etc. 5. The multicollision technique from Joux allows an attacker to discover 2k messages with the same hash for an n-bit iterated hash function, using only about k 2n/2 compression functions of work. This technique can be used to make a set of 2k expandable messages which all yield the same hash output. The full power of combining these techniques remains to be investigated. 4 Using Expandable Messages to Find Second Preimages
An n-bit hash function is supposed to resist second preimage attacks up to about 2n work. That is, given one message M, the attacker ought to have to spend about 2n work to nd another message that has the same hash value as output.
4.1 The Long Message Attack
Here is a general (and previously known) way to violate the second-preimage resistance of a hash function without Damg ard-Merklestrengthening [MvOV96]: Start with an extremely long message of 2R + 1 blocks. An attacker who wishes to nd another message that hashes to the same value with a 160-bit hash function can do so by nding a message block Mlink such that, from the IV of the hash, h[ 1], h = F (h[ 1],Mlink) yields a value h that matches one of the intermediate values of the hash function in processing the long message. Since the message has 2R such intermediate values, the attacker expects to need to try only about 2160 R message blocks to get a match. That is, when R = 64, the attacker has 264 available target values, so each message block he tries has about a 2 96 chance of yielding the same hash output as some intermediate hash value from the target message. The result is a shorter message, which has the same hash output up until the nal block is processed. The length padding at the end of the Damg ard-Merkleconstruction foils this attack. Note that in the above situation, the attacker has a message that is shorter than the 255-block target message, which leads to the same intermediate hash value . But now, the last block has a di erent length eld, and so the attack fails—the attacker can nd something that’s almost a second preimage, but the length block changes, and so the nal hash output is di erent.
4.2 Long-Message Attacks with Expandable Messages
Using expandable messages, we can bypass this defense, and carry out a second- preimage attack despite the length block at the end. This attack was rst discov- ered by Dean [Dea99]. We start with a long message as our target for a second preimage, nd an expandable message which will provide messages over a wide range of lengths, and then carry out the long-message attack from the end of that expandable message. We then expand the expandable message to make up for all the message blocks that were skipped by the long message attack, yielding a new message of the same length as the target message, with the same hash value.
ALGORITHM: LongMessageAttack(Mtarget) Find the second preimage for a message of 2k + k + 1 blocks.
Variables: 1. Mtarget = the message for which a second preimage is to be found. 2. Mlink = a message block used to link the expandable message to some point in the target message’s sequence of intermediate hash values. 3. A = a list of intermediate hash values 4. hexp = intermediate chaining value from processing an expandable mes- sage. Steps: 1. C = MakeExpandableMessage(k) 2. hexp = the intermediate hash value after processing the expandable mes- sage in C. 3. Compute the intermediate hash values for Mtarget: – h[ 1] = the IV for the hash function – m[i] = the ith message block of Mtarget. – h[i] = F (h[i 1], m[i]), the ith intermediate hash output block. Note that h will be organized in some searchable structure for the attack, such as a hash table, and that elements h[0, 1, ..., k] are excluded from the hash table, since the expandable message cannot be made short enough to accommodate them in the attack. 4. Find a message block that links the expandable message to one of the intermediate hash values for the target message after the kth block. – Try linking messages Mlink until F (hexp,Mlink) = h[j] for some k + 1 j 2k + k + 1. 5. Use the expandable message to produce a message M that is j 1 blocks long. k 6. Return second preimage M ||Mlink||m[j + 1]||m[j + 2]...m[2 + k + 1] (if j = 2k + k + 1, then no original message blocks are included in the second preimage). Work: The total work done is the work to nd the expandable message plus the work to nd the linking message. 1. For the generic expandable message- nding algorithm, this is k 2n/2+1+ 2n k+1 compression function calls. 2. For the xed-point expandable message- nding algorithm, this is 3 2n/2+1 + 2n k+1
The longer the target message, the more e cient the attack relative to a brute-force preimage search, until the search for the expandable message be- comes more expensive than the long-message attack. For SHA1 and SHA256, the maximum allowed message length is 264 1 bits, which translates to about 255 512-bit blocks of message. For SHA384 and SHA512, the maximum allowed message length is 2128 1 bits, which translates to about 2118 1024-bit blocks of message. Let 2R be the maximum number of message blocks allowed by the hash function. The total work of the generic expandable-message form of the attack is then R 2n/2+1 + 2n R+1 compression function calls.
An Illustration. To illustrate this, consider a second preimage attack on the RIPEMD-160 hash function [DBP96]. The longest possible message for RIPEMD-160 is 264 1 bits, which translates into just under 255 blocks. For simplicity, we will assume the target message is 254 + 54 + 1 message blocks (about 260 bytes) long.
1. Receive the target message and compute and store all the intermediate hash values. 2. Produce a (1, 54 + 254)-expandable message. This requires about 54 281 compression function computations. 3. Starting from the end of the expandable message, we try about 2106 di erent message blocks, until we nd one whose hash output is the same as one of the last 54 + 254 intermediate hash values of the target message. This requires computing about 2106 compression functions on average. 4. Expand the expandable message to compensate for the message blocks of the target message skipped over, and thus produce a second preimage. This takes very little time.
Summary of the Attack. The long-message attack can be summarized as follows: For a target message substantially less than 2n/2 blocks in length, the work is dominated by the long message attack. Thus, a second preimage attack on a 2k-block message takes about 2n k+1 compression function computations, assuming abundant memory.
4.3 Variations on the Attack
Some straightforward variations of this attack are also possible, drawing from the variations available to the expandable messages. For example, the algorithms for producing an expandable message work from any starting hash value, and are not a ected by the message blocks that come after the expanded message. Thus, this attack can be used to “splice together” two very long messages, with an expandable part in the middle. Similarly, if it is important that the second preimage message start with the same rst few hundred or thousand message blocks as the target message, or end with the same last few hundred or thou- sand blocks, this can easily be accommodated in the attack. Another variation is available by using Joux’s multicollision- nding trick, or the related ones de- scribed below: By setting up the expandable message to be a 2u-multicollision, we can nd 2u distinct second preimages for a given long message, without adding substantial cost to the attack. Additionally, keyed constructions that leave the attacker with o ine collision search abilities are vulnerable to the attack; for example, the “su x mac” construction [MvOV96], MACK (X) = Hash(X||K) is vulnerable to a second preimage attack, as well as the much more practical, previously-known collision attack.
Low-Memory and Parallel Versions of the Attack. These methods for nding expandable messages assume unlimited memory. In the real world, mem- ory is limited, and bandwidth between processing units and memory units is likewise limited. This doesn’t raise a di culty to the attack. For n-bit hash functions whose maximum input size in message blocks is substantially less than 2n/2, the parallel collision search techniques of [vOW96,vOW99] allow both our generic attack and the xed-point attack of Dean to go forward at approximately the stated cost; the search for a linking message (the long message attack) dom- inates the work.
4.4 The Attack in Perspective Our attack allows the nding of a second preimage on a 2k block long target message with a certainty of success. A previously known attack originally noted by Merkle is somewhat similar in spirit [MvOV96,Pre05]: an attacker is given 2k candidate target messages, and nds one preimage with 2n k work. While that attack wasn’t able to nd a second preimage for a speci c desired message, it makes our result and the earlier result of Dean somewhat less surprising. It is worth noting that Merkle’s result applies to any n-bit hash function, even one constructed from random oracles. Our attack di ers from that of Dean only in its universality–Dean’s attack applies only to hash functions whose compression functions allow easy nding of second preimages, whereas ours apply to any iterated hash function with an n-bit intermediate hash value.
5 Expandable Messages and Multicollisions
In [Jou04], Joux demonstrates a beautiful way to produce a large number of messages that collide for an iterated hash function, with only a little more work than is needed to nd a single pair of messages that collide. Here, we demonstrate ways to use expandable messages to nd multicollisions, and ways to combine the Joux technique with expandable messages to add exibility to the structure of the multicollisions. We construct a multicollision by concatenating two or more expandable mes- sages, and then varying the length of each so that the sum of their lengths stays the same. For example, if we concatenate a (1,1024)-expandable message with another (1,1024)-expandable message, we get a 1024-collision of 1025 block long messages. By concatenating a large number of such messages, we can get a much larger multicollision.
5.1 Multicollisions Using Fixed Points Using xed-point expandable messages, multicollisions which are much cheaper than those found by Joux are available. Recall that for an n-bit hash func- tion, nding a xed-point expandable message which is expandable up to the maximum message length of the hash function costs about 2n/2+1 compression function computations. Consider a 160-bit hash function with a maximum of 255 message blocks. Now, a very simple 255-collision is available for about 282 = 4 280 work, as opposed to 55 280 work–this is constructed by concatenating two xed-point expandable messages, and always making the sum of their lengths 255 blocks. Concatenating three such expandable messages produces a 2107-collision, and so on, following the rule that a multicollision consisting of K expandable mes- R sages in a hash function with a maximum length of R blocks produces K 1 - multicollisions with about K 281 work. These multicollisions are of unreasonable length, but they’re generally cheaper than Joux’ multicollisions. At more reasonable lengths, they’re still interesting, but they become more expensive than Joux’ multicollisions. For example, con- catenating ten expandable messages together and limiting message length to 1034 blocks total, we get about a 280-multicollision using this technique; for the same cost, Joux’ technique would give a 21024-multicollision.
5.2 Using Generic Expandable Messages
The cost of nding a single xed-point expandable message is within a factor of two of the cost of nding a single collision in Joux’ scheme. The cost of nding a generic (1,K)-expandable message is about lg(K) 2n/2. This means that in general, generic expandable messages cannot be used to make multicollisions cheaper than those of Joux.
5.3 Combining With Joux
Finally, it is possible to combine Joux multicollisions with expandable-message multicollisions. This allows multicollisions to be constructed that look quite dif- ferent from the Joux multicollisions, and are somewhat more exible in structure. This may allow Joux attacks to go forward even on cascaded constructions that attempt to foil his attack. As an example, a multicollision may be formed by alternating (1, 2)-expandable messages and individual collisions as sought by Joux’ method, with a nal (10, 1024)-expandable message at the end. This permits the individual colliding message blocks to appear at di erent positions in di erent messages, without altering the nal hash value.
6 Conclusions and Open Questions
In this paper, we have described a generic way to carry out long-message second preimage attacks, despite the Damg ard-Merklestrengthening done on all modern hash functions. These attacks are theoretical because 1) they require more work than is nec- essary to nd collisions on the underlying hash functions, and 2) the messages for which second preimages may be found are generally impractically long. However, they demonstrate some new lessons about hash function design: 1. An n-bit iterated hash function provides fundamentally di erent security properties than a random oracle with an n-bit output. This was demon- strated in one way by Joux in [Jou04], and by another here. 2. An n-bit iterated hash function begins to show some surprising properties as soon as an attacker can do the work necessary to nd collisions in the underlying compression function. 3. An n-bit iterated hash function cannot support second-preimage resistance at the n-bit security level, as previously expected, for long messages. 4. Easily found xed points in compression functions (such as those based on the Davies-Meyer construction) allow an even more powerful second- preimage attack described in [Dea99]. The important lesson here is that the standard construction of iterated hashes from Merkle and Damg arddoes not provide all the protection we might expect against attackers that can do more than 2n/2 compression function com- putations. In some sense, the hash function is “brittle,” and begins to lose its claimed security properties very quickly once the attacker can violate its collision resistance by brute force. We believe these results, when combined with those of Dean and Joux, re- quire a rethinking of what security properties are expected of an iterative hash function with an n-bit intermediate state. We see three sensible directions for this rethinking to take: 1. A widespread consensus that an n-bit iterated hash function should never be expected to resist attacks requiring more than 2n/2 operations. This would invalidate current uses of hash functions in cryptographic random- number generation, as in [KSF99,DHL02,Bal98], key derivation functions as described in [AKMZ04,NIST03,X963], and many other applications, and seems the least palatable outcome. 2. A clear theoretical treatment of the limits that exist for n-bit hash func- tions, and precisely what attacks more demanding than collision search they may be expected to resist. (For example, none of these recent results appear to be applicable when the attacker cannot do o ine collision search. Simi- larly, these attacks do not apply when only a single message block is being processed. Perhaps these observations can be formalized.) 3. New constructions for hash-function round functions. For example, XORing in a monotomic counter as part of the round function would resist the attacks in this paper. 4. New constructions for hash functions in the vein of [Luc04], which maintain much more than n bits of intermediate state in order to make collision attacks on intermediate states harder (require 2n work). We believe that the region between 2n/2 and 2n is a rich area for the crypt- analysis of iterated hash functions, and expect to see other research results in the future. Absent a solid theoretical treatment of the security properties of n- bit iterative hashes along the lines of [PGV93] and [BRS02], expanded to deal thoroughly with the full hash construction, at this point it is di cult to justify using them in applications requiring more than n/2 bits of security for messages longer than one block with any con dence. We hope this work spurs such a treatment, as well as further cryptanalysis.
7 Acknowledgements
The authors wish to thank Bill Burr, Morris Dworkin, Matt Fanto, Niels Fergu- son, Phil Hawkes, Julie Kelsey, Ulrich K uhn,Arjen Lenstra, Stefan Lucks, Bart Preneel, Vincent Rijmen, David Wagner, and the anonymous referees for many useful comments and discussions on the results in this paper.
References
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A Finding Fixed Points E ciently in Many Compression Functions
Finding xed points in many hash compression functions is simple.
A.1 Davies-Meyer
Most widely used hash functions have compression functions designed around very large block-cipher-like constructions, following the general Davies-Meyer model. For the SHA and MD4/MD5 families, as well as Tiger, if E(K,X) is a very wide block cipher, with K the key and X the value being encrypted, then the compression function is:
F (H,M) = E(M,H) + H for some group operation “+”. For these compression functions, it is possible to compute the inverse of this block-cipher-like construction, which we can denote as E 1(K,X). This makes it possible to nd xed points in a simple way, as discussed in [MOI91] and [PGV93]:
1. Select a message M. 2. Compute H = E 1(M, 0). 3. The result gives a xed point: F (H,M) = H. A property of this method for nding xed points is that the attacker is able to choose the message, but he has no control whatsoever over the hash value that is a xed point for a given message. Also note that for these hash functions, each message block has exactly one xed point.
A.2 Snefru
Snefru is derived from a block-cipher-like operation that operates on a much larger block than the hash output, and which e ectively has a xed “key.” Let E(X) be this xed “encryption” of a block. Further, let n be the hash block size, m be the message block size, lsbn(X) be the least signi cant n bits of X, and msbn(X) be the most signi cant n bits of X. Note that E(X) operates on n + m-bit blocks. The compression function is derived from E(X):
F (H,M) = lsbn(E(H||M)) + H where the hash input and output are each n bits wide, and where lsbx(Y ) rep- resents the least signi cant x bits of the value Y . We can nd xed points for Snefru-like compression functions as follows, letting E 1(X) be the inverse of E(X) once again:
1. Choose any X whose least signi cant n bits are 0. 2. Compute Y = E 1(X). 3. Let H = msbn(Y ) and M = lsbm(Y ). 4. The result gives a xed point: F (H,M) = H.
This method gives the attacker no control over the message block. Unlike the Davies-Meyer construction, there is no guarantee that a given message block has even one xed point; we would expect for some message blocks to have many, and for others to have none. Note that the Snefru construction could easily be altered to make xed points very hard to nd, when the size of the message and hash blocks are equal, by the compression function as:
F (H,M) = lsbn(E(H||M)) + H + M or F (H,M) = lsbn(E(H||M)) + H + M + msbm(E(H||M)) Also note that many other compression function constructions, such as the Miyaguchi-Preneel construction used by Whirlpool and N-Hash and the con- struction used by RIPEMD and RIPEMD-160, do not appear to permit a generic method for nding xed points.